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The question asks to distribute the first 10 even positive numbers (2, 4, 6, 8, 10, 12, 14, 16, 18, and 20) into the vertices of a regular pentagon, with each side of the pentagon containing two numbers. The goal is to ensure that the sum of the numbers on each side of the pentagon is the same and minimal.

### Steps:
1. **Numbers to use:** The even numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20.
2. **Arrange in a pentagon:** There are 5 sides, and each side must have two numbers.
3. **Target condition:** The sum on each side of the pentagon should be equal and as small as possible.

#### Analysis:
The sum of the numbers from 2 to 20 (even numbers) is:
\[
2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = 110
\]
If we divide the 10 numbers into 5 groups (sides of the pentagon), each side should have two numbers, and the total sum for each side is:
\[
\frac{110}{5} = 22
\]
So, the sum of the two numbers on each side of the pentagon must be 22.

### Arranging the numbers:
We now need to select pairs of numbers from 2 to 20 that sum to 22:
- \(2 + 20 = 22\)
- \(4 + 18 = 22\)
- \(6 + 16 = 22\)
- \(8 + 14 = 22\)
- \(10 + 12 = 22\)

This arrangement ensures that the sum on each side of the pentagon is 22, which is the minimum possible sum.

### Identifying the numbers next to 20:
From the pairs, we see that the number 20 is paired with 2. So, 20 is adjacent to 2 and 18.

### Total sum of adjacent numbers to 20:
The numbers adjacent to 20 are 2 and 18. Their sum is:
\[
2 + 18 = 20
\]

### Answer:
The correct answer is \(20\).

---

Would you like further clarification on how to approach similar problems? Here are five related questions to consider:
1. How can you generalize this method for different polygons with even-numbered sets?
2. What would change if odd numbers were used instead of even numbers?
3. Can you use this method for irregular polygons?
4. How would the approach change if the sum of the numbers on each side had to be a specific value (not minimal)?
5. What are other applications of these types of combinatorial arrangements?

**Tip:** When solving problems involving geometric shapes and numbers, look for symmetries and constraints to simplify the arrangement process.

En el siguiente pentágono regular, distribuya los 10 primeros números pares positivos, uno por cada casilla circular, de manera que la suma de los números ubicados en cada lado del pentágono sea la misma y la mínima posible. Indique junto y entre qué números se encuentra el número 20. Dé como respuesta la suma de dichas cantidades.

Learn how to distribute the first 10 even positive numbers (2 to 20) across the vertices of a regular pentagon. The goal is to ensure that the sum on each side is equal and minimal. Detailed analysis of number pairing and combinatorics is included. Suitable for Grades 9-11.

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